Intervals

Intervals are the raw material from which melodies are created, and provide the building blocks of harmony. They are the musician's tool to control feeling and tell a story.

This section of ten web pages explores intervals from the very basics of vibration to the terminology used in music theory. A general reference for these topics is W. A. Mathieu's Harmonic Experience ([Mathieu 1997]).

What is Sound?

Sound is a fluctuation in the air pressure — an area where air particles are more densely packed or more loosely packed than the surrounding air.
Unlike the pressure changes brought about by weather (e.g. “a low-pressure front bringing in moisture in the early evening …”), sound involves local, relatively rapid change in air pressure.
Because air is a gas, air particles are free to move about and bump into neighboring air particles.
Local changes in air pressure tend to spread out through the air
with the character of a wave motion that is triggered as the air particles interact
with their neighbors ([Everest 2001]).
This domino effect tends to move through the air at the speed of sound — about 760 miles per hour or one mile every five seconds at typical conditions ([Dean 1979]).

Vibration

We hear sounds because our ears respond to changes in air pressure.
If the change in air pressure oscialates — high pressure / low pressure / high pressure / low pressure
— our ears, nervous system, and brain convert changes in air pressure into a perception of sound.

If you strike the drum head, it vibrates and pushed the air into waves of high pressure (more tightly packed air molecules) and low pressure (less tightly packed air molecules). Those waves of alternating pressure
are called air pressure waves.
They affect the neighboring air and radiate out from the source, with reducing pressure and precision. When it reaches our ears, it is perceived as sound.

So air molecules do not move from the intrument
to the ear — we would feel that as wind. Each molecule vibrates back and forth in a limited region. This understanding came fairly recently — Jean Baptiste Joseph Fourier (1768–1830) came to understand this principle and then to map out mathematically the frequencies and sine waves involved in sound ([Fourier 1888]).

In the example above, the vibrations of a drum head die out quickly and we perceive a brief sound. If the sound source continues to produce air pressure waves, we perceive a long sound. And if the oscillation happens at a steady rate, we perceive a steady tone or pitch or note.

The picture above is often represented as a graph of the changing air pressure over time:

The blue line represents how the air pressure changes over time. The distance in time between two neighboring troughs in the graph (the low-pressure points) or two neighboring peaks in the graph (the high-pressure points) is called one cycle. If the difference in pressure between the low- and high-pressure points is greater, the blue line would appear taller and we would perceive a louder sound.

When people say that sound is vibrational energy, one aspect of what they are talking about is the energy that is transferred from the drum head to your ear, in waves of air pressure.

Flute Vibrations

It's easy to see how the head of a bass drum shown above causes air pressure waves. As the drum head vibrates back and forth, the air molecules are pushed and pulled into areas of higher pressure and lower pressure.

The string on a stringed instrument has a similar effect on the air. A guitar string vibrates when plucked by a finger or a pick, a piano string vibrates when struck by a hammer inside the piano, and a violin string continuously vibrates when vibrated by the coarse hairs of a bow drawn across it.

But what about a flute?

How flutes cause air pressure waves is best shown by these moving images from
Luchtwervels in een blokfluit «Air Vortices in a Recorder»
([Hirschberg 1999]). They are both images of how the air coming out of the flue crosses the sound hole and hits the splitting edge of a recorder:

Airstream at the sound hole of a recorder

The left image shows the behavior of the airstream coming out of the flue at the onset of a tone - what would be the attack at the start of a note. The air initially flows up and away without any vibration or osciallting pattern.

The right image shows what happens a bit later, after an oscillation has been established. This osciallation happens because of the specific shape of all the aspects of the flute, but in particular the shape of the flue, the sound hole, the splitting edge, and the flute's sound chamber.

Oscillations of the air jet in the mouth of a recorder with a fundamental frequency of 513 Hz.
The core gap is 1 mm, the distance between the core and the output gap of the labium is 4 mm.
The visualization is obtained using the so-called Schlieren technique: the whistle blows CO2
and creates a contrast in the refractive index.

Presumably, the “core” referred to is the height of the flue.

So the air pressure wave is created in flutes by an oscillation of air above and below the splitting edge.

Frequency

The number of full cycles of oscillation (corresponding to the number of peaks in graph of the vibration) that happen every second is typically called the frequency. Frequency is usually measured in the number of "cycles per second" or "Hertz" (abbreviated "Hz") after the German physicist Heinrich Hertz. So if there are 100 peaks in air pressure each second, the sound is said to have a frequency of 100 Hz.

Our ears are designed to hear sounds with frequencies of between roughly 25 and 15,000 Hertz.

Pitch

Instruments that produce air pressure waves that have a steady frequency are said to be "pitched instruments", because the sound we hear has a recognizable tone or pitch. All flutes are pitched instruments.

Its common to hear physicists talk about frequencies in terms of Hertz, but musicians tend to use a system of musical note names that have been developed. The musical notes better represent how we hear music, but they also make it easy to forget the vibrational basis of our music. When something in music says "A=440", it's a shorthand for saying that the musical note named "A" corresponds to 440 Hertz.